Boundary-Layer Meteorology

, Volume 8, Issue 3–4, pp 401–417 | Cite as

Analysis of atmospheric flow over a surface protrusion using the turbulence kinetic energy equation

  • Walter Frost
  • W. L. Harper
  • G. H. Fichtl
Article

Abstract

Flow over surface obstructions can produce significantly large wind shears such that adverse flying conditions can occur for aeronautical systems (helicopters, STOL vehicles, etc.) Atmospheric flow fields resulting from a semi-elliptical surface obstruction in an otherwise horizontally homogeneous statistically stationary flow are modelled with the boundary-layer / Boussinesq-approximation of the governing equation of fluid mechanics. The turbulence kinetic energy equation is used to determine the dissipative effects of turbulent shear on the mean flow. Mean-flow results are compared with those given in a previous paper where the same problem was attacked using a Prandtl mixing-length hypothesis. The diffusion and convection of turbulence kinetic energy not accounted for in the Prandtl mixing-length concept cause departures of the mean wind profiles from those previously computed, primarily in the regions of strong pressure gradients.

Iso-lines of turbulence kinetic energy and turbulence intensity are plotted in the plane of the flow. They highlight regions of high turbulence intensity in the stagnation zone and sharp gradients in intensity along the transition from adverse to favourable pressure gradient.

Keywords

Turbulence Kinetic Energy Turbulence Intensity Wind Shear Wind Profile Turbulent Shear 

Nomenclature

a1

proportionality constant of shear stress to turbulence kinetic energy

b

height of elliptical body

E0

reference turbulence kinetic energy

e

dimensionless turbulence kinetic energy (e*/E0)

G

function ofz*/δ

k

effective viscosity

L

mixing length

l

Prandtl mixing length

P

dimensionless pressure (P*/ϱUϱ)

Sce

ratio of turbulent eddy viscosity to a diffusion coefficient for energy transfer

u

dimensionless velocity component inx direction (u*/U)

u*

friction velocity,\(\sqrt {\tau /\varrho }\)

û

turbulence intensity

Ue

dimensionless velocity along streamline (U* e /U)

U

reference velocity

υ

dimensionless velocity component iny direction (υ*/U)

w

dimensionless velocity component in z direction (w*/U)

x

dimensionless horizontal coordinate parallel to direction of flow (x*/b)

z

dimensionless vertical coordinate (z*/b)

α

aspect ratio of ellipse

γ

intermittency

δ

boundary-layer thickness

ε

dissipation term of turbulence kinetic energy equation

κ

von Kármán's constant

λ

length of separation bubble

υ

kinematic viscosity

ϱ

mean flow density

σu

root-mean-square value of fluctuatingu velocity component

σV

root-mean-square value of fluctuatingυ velocity component

σw

root-mean-square value of fluctuatingw velocity component

Τ

turbulent shear stress

υ

convection velocity

Superscripts

*

dimensional quantity

turbulent fluctuating component

¯()

ensemble average operator

Subscripts

0

initial value from undisturbed upstream flow

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bradshaw, P., Ferriss, D. H., and Atwell, N. P.: 1967, ‘Calculation of Boundary Layer Development Using the Turbulent Energy Equation’,J. Fluid Mech. 28, 593–616.Google Scholar
  2. Byrne, William M., Jr.: 1970, ‘Use of the Turbulence Kinetic Energy Equation in Prediction of Non-equilibrium Turbulent Boundary Layers’, Ph.D. Thesis, University of Missouri at Rolla.Google Scholar
  3. Byrne, William M., and Lee, S.C.: 1972, ‘A Differential Method for the Prediction of the Effects of Atmospheric Boundary Layer Turbulence Using the Turbulence Kinetic Energy Equation’,Proceedings of the Symposium on Air Pollution. Turbulence and Diffusion, (held at New Mexico State University, Las Cruces, December 7–10, 1971), Publ.Atmospheric Fluid Dynamics Division, Sandrei Lab., Albuquerque, N.M., H. W. Church and R. E. Luna (eds.), pp. 231–243.Google Scholar
  4. Dutton, J. A. and Fichtl, 1969, ‘Approximation Equations of Motion for Gases and Liquids’,J. Atmospheric Sci. 26, 241–254.Google Scholar
  5. Fichtl, G. H.: 1973, ‘Problems in the Simulation of Atmospheric Boundary Layer Flows’,AGARD Conference Proceedings No. 140.Google Scholar
  6. Frost, W. and Harper, W. L.: 1974, ‘Analysis of Atmospheric Flow over a Surface Protrusion Using the Turbulence Kinetic Energy Equation with Reference to V/STOL Flight Dynamics’, report in preparation.Google Scholar
  7. Frost, W., Maus, J. R., and Fichtl, G. H.: 1974, ‘A Boundary Layer Analysis of Atmospheric Motion over a Surface Obstruction’,Boundary-Layer Meteorol. 1, 197–216.Google Scholar
  8. Frost, W., Maus, J. R., and Simpson, W. R.: 1973, ‘A Boundary Layer Approach to the Analysis of Atmospheric Motion over a Surface Obstruction’, NASA CR-2182.Google Scholar
  9. Harsha, P. T.: 1970, ‘Free Turbulent Mixing: A Critical Evaluation of Theory and Experiment’, Ph.D. Thesis, University of Tennessee.Google Scholar
  10. Jackson, P. S. and Hunt, J. C. R.: 1974, ‘Turbulent Wind Flow over a Low Hill,J. Fluid Mech. 64, 529–563.Google Scholar
  11. Launder, B. E. and Spalding, D. B.: 1972,Mathematical Models of Turbulence, Academic Press, London.Google Scholar
  12. Lee, S. C. and Harsha, P. T.: 1970, ‘The Use of Turbulence Kinetic Energy in Free Mixing Studies’,AZAA J. 8, 1026–1032.Google Scholar
  13. Robertson, J. M. and Taulbee, D. B.: 1969, ‘Turbulent Boundary Layer and Separation Flow Ahead of a Step’, in H. J. Weisset al. (eds.),Developments in Mechanics, No. 5, Iowa State Univ., Iowa.Google Scholar

Copyright information

© D. Reidel Publishing Company 1975

Authors and Affiliations

  • Walter Frost
    • 1
  • W. L. Harper
    • 1
  • G. H. Fichtl
    • 2
  1. 1.Space InstituteUniversity of TennesseeTullahomaUSA
  2. 2.NASA Marshall Space Flight CenterHuntsvilleUSA

Personalised recommendations